Brauer morphism between modular Hecke algebras

JOURNAL

OF ALGEBRA

115,

1-31 (1988)

Brauer Morphism

between

Modular

Hecke Algebras

MARC CABANES Equipe de la ThPorie des Groupes Finis, Ecole Normale SupPrieure DMI, 75230 Paris Cedex 05, France Communicated by Walter Feir

ReceivedDecember19, 1984

INTRODUCTION

Noticeable progress in the knowledge of modular representations of finite groups has been accomplished by considering (papers of Broue, Puig and Scott) the most general situations where one can define the relative traces ideal and the Brauer morphism (see [4]). Our aim in this paper is to describe what happens in the case of p-permutation modules and certain derived modules (most of our matter comes from unpublished results of L. Puig), then to apply that to representations of split BN-pairs in natural characteristic. These two subjects have been treated previously in two papers. The first one, by Scott [26], points out the possibility of setting up for permutation modules a theory analogous to Brauer’s for group algebra and which contains it; the results he obtains are analogues of Brauer’s first and second main theorems and Brauer’s theorems on defect 0 and 1. The other paper, by Tinberg [29], presents the results of Curtis and Richen and Sawada and Green on irreducible modules for split &V-pairs in natural characteristic and certain associated indecomposable modules. Tinberg’s paper contains two further results: an induction formula (leading to very precise information about the dimension of the involved modules) and a computation of vertex, the latter result using Scott’s paper. Our study of the link between these subjects permits us first to simplify (Part B) several proofs of the Curtis-Richen-Sawada theory (existence of “weights,” structure coefficients for the modular “Hecke algebra” End,,(indG, k)) and after that to obtain certain results about indecomposable direct summands of indG,k (Part C). In particular we check that irreducible kc-modules (G is a split BN-pair represented in natural characteristic) are in l-l correspondence with G-conjugacy classesof pairs (I’, rr), where V is a p-subgroup of G and 7can irreducible kJlr,( V)/V-module of defect 0. 0021-8693/88 63.00 Copyright cl 1988 by Academic Press, Inc All rights of reproduction in any form reserved.

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MARC CABANES

We also show that the question “which are the p-subgroups V of G such that JQ V)/V has a block of p-defect 0’7 can be answered without the use of a Borel-Tits result. We end by giving a computation that links the summands of indz k with the summands of the same module for a Levi subgroup. This paper can be considered as trying to make precise the link between the representations of G and the ones of its parabolic-or Levi-subgroups (Levi subgroups being a very systematic way of imbedding certain finite simple groups in other ones); this is done by means of Brauer morphism between modular Hecke algebras under an explicit form (B.8, Theorem 6) coming from a general result (A.3, Proposition 3) on permutation modules. Certain applications of this paper are in the exposition of CurtisRichen-Sawada theory on classification of irrducible representations of split BN-pairs, so we have tried to make it self-contained; in particular we have given an account of the main known properties of the involved representations (our B.6). Part A deals with permutation representations, the Brauer morphism and some related results, most of them due to L. Puig (see also [26]). Part B gives the Green-Sawada theory of modular irreducible representations of split BN-pairs; this part is independent of Part A except for some examples treated in Part A. In Part C we apply the results of Part A to the framework set in Part B. Background Notations and Results

If G is a group and H a subgroup of G, we denote by h&(H) the normalizer of H in G, set -P’&(H) = ,#‘i(H)/H, and let G&(H) denote the centralizer of H in G. If k is an algebraically closed field, and A a k-algebra of finite dimension, a k-vector space M is said to be an A-module if, and only if, it has a finite dimension over k and is an A-module in the usual sense. The notions of morphism, isomorphism, irreducibility (or simplicity), semi-simplicity, socle (the biggest semi-simple submodule of M, denoted by sot M), head (the biggest semi-simple quotient denoted by hd M), direct summand, indecomposability, etc., are the usual ones. One knows that every decomposition of M as a direct sum of indecomposable submodules involves the same number of summands (we call this number the “width” of M, denoted wth(M)), and the same number of summands in any given isomorphism class of indecomposable modules. If A is a finite dimensional k-algebra we use the usual notion of idempotents. If ei, ez are two idempotents of A we say that: -

“ei is orthogonal to e, ” if, and only if, elez =ezel =O; “ez contains e,,” denoted e, ce2, if, and only if, e,e2 =ezel =e,.

BRAUER

MORPHISM

3

An idempotent is said to be primitive if and only if it contains exactly two idempotents: 0 and itself. There is a bijection between the conjugacy classes of primitive idempotents and the conjugacy classes of maximal left ideals of A: if i is a primitive idempotent of A there exists a unique (up to conjugation) maximal left ideal 4, such that i# Ji. Conversely, if ,I is a maximal left ideal the primitive idempotents of A which are in A\.,&’ form a conjugacy class. The corresponding bijection between conjugacy classes of primitive idempotents and irreducible representations of A is obvious, and if i is a primitive idempotent, the k-dimension of the associated irreducible representation of A equals the number of conjugates of i in a decomposition of the unity of A (see [ 111). If E is a finite subset of A we denote by YE the element C,, E x of A. We consider kG as a k-algebra. If A4 is a kG module and His a subgroup of G we define the restriction of M to H, res,M, in the usual way; we denote the fixed points by M”, this subspace being a k&L(H)-module. If Hc H’ c G we define the relative trace, a map from MH to Mw, as Trz’(x) = YE. x for x E MH and E a representative system of HI/H. One has the Mackey formula: if K is a subgroup of G and x E MH, Tri’(x) = .x). The image of the map Tr$ is denoted ME’ (see c KgHcH” T~~n~Hk [ 13, pp. 87-921). If N is a kH-module we define the induction indz M = kG OkH M in the usual way. We have the Mackey formula: resk indg M = xkRHc Gindi,,, (g @ M) corresponding to the decomposition kGOkH N = QKgH c, EKgHx @ N. If A is a k-algebra we say that A is a G-algebra if G acts on A by k-algebra automorphisms; then A:’ is a two-sided ideal of the ,&(H’)algebra A”‘.

If M is a kG-module, End, M is a G-algebra (for conjugation); we denote End,,.(M) = (End, M)“‘), with the well-known correspondences: between direct summands of M and idempotents of End,, M, and between indecomposable direct summands of ii4 and primitive idempotents. If M is an indecomposable kc-module there exists a set of p-subgroups of G which we will denote vtx(M) such that (see [15]): VV~vtx(iU) there exists NV, a kV-module, such that MJ ind$ N,, with V minimal as such. This set vtx(M) is a conjugacy class. The elements of vtx(M) are called the vertices of M; if I” is a subgroup of G we denote v’c vtx(M) for the condition: 3 VE vtx(M) v’ c V. If V is a subgroup of G, the condition: 3NV a kV-module such that Ml indG,N, is equivalent to 1 E (End, M)Gy. The indecomposable k V-modules NV for VE vtx(M) and M 1indG,N, are called the sources of M. If H is a subgroup of G and N is a KH-module such that MI ind$ N then H contains a vertex V of M and resy N admits a

4

MARCCABANES

source as direct summand (for the subject of vertices and sources see [ 13, pp. 111-123; 15; 111). If I E Mor(G, k*) we will often denote by 1 the associated one-dimensional kc-module. One denotes by X,(G) the set of (isomorphism classes) of one-dimensional kG-modules. PART A. p-PERMUTATION MODULES AND THE BRAUER MORPHISM A.1. BRAUER MORPHISM A.l.l. Let G be a finite group, H a subgroup of G, k a field and M a kc-module. One denotes by 9”(M) the k-submodule CHzc H Mz., where the sum is taken over all the subgroups of H different from H, with the rule that 9’(M) = 0. Then 9”(M) is a kM(/,(H)-submodule of MH. One denotes by M(H) the J&(H)-module MH/9”(M) and Br, the surjective J1/;;(H)-morphism from MH onto M(H). As an immediate consequence of the Mackey formula we have Br, 0Trz = Trih(H)o Br, as a map defined on M”. In particular Br,(Mg) = M(H)$~(“) (see [4, 51). Moreover if M = M, Oc Mz one has M(H) = M,(H) O.,tGcHl M,(H). A.1.2. If M is a G-algebra, yH(M) is a two-sided ideal in the Jlr,(H)algebra MH and Br, is an JZrc(H)-algebra morphism.

A.2. ~-PERMUTATION MODULES Starting from the notion of a permutation module it seemsquite natural to consider the modules that are direct summands of permutation modules. A.2.1. The definition we give proves handiest. To check that it coincides with direct summands of permutation modules or with modules containing a stable basis under a Sylow p-subgroup, see [3,0.4]. DEFINITION 1. If G is a finite group, k a field of finite characteristic p, and Y a kG-module, one says that Y is a p-permutation module if, and only if, it is a direct sum of indecomposable submodules each of them being of source 1.

Clearly, that class of modules is stable by direct sum, direct summand and restriction.

BRAUERMORPHISM

5

A.2.2. A Decomposition PROPOSITION 1. If Y is a p-permutation kc-module and V is a normal p-subgroup of G, one has the decomposition

y= y, oc y,, with

(i) Y?(V)=O, (ii) (iii) (iv)

Y,V= Y, , 4 “‘(Y) = Yr, .a”‘(End, Y) = (xe End,, Y; xY, c Y2}.

Furthermore, Y(V) N Y, as kG-modules. Proof: Let us choose a decomposition of Y as a direct sum of indecomposable kG-modules, and let us denote by Y, the sum of those terms whose vertex contains V (then (ii) is satisfied since Vu G), Y, being the sum of the others. One has Y= Y,@ Y,, and (i) follows from [3, 3.2(l)]. Assertion (iii) comes from (i) and (ii). (iv)

Let us set A =: End, Y.

s”(A)c {XE A”; xY,cY,}:IfV’$ V,xxAAandyEY,,onehasto prove that TrL.(x) .v c Y,. Since y E Y” then Trr.(x) .y = TrL(x .y) with x.y in Y”‘. But now TrL,(x.Y)E Y::,c4Y(Y)=~a(Y,)+s”(Y,)c Y, because 9 “‘( Y, ) = 0. (x E A “; xY, c Y?} c 9 ‘(A): Let 1 = e, + e, be the decomposition of the unity of AG associated with the decomposition Y= Y, @ Y,. Let e2= LI i be the decomposition of e, associated with a decomposition of .YZas a direct sum of indecomposable kG-modules whose vertices do not contain V. Then, let Vi be a vertex of iY,: V, 5 G V and iE A$,. But, if iE Z, Az,c.T’(A): A~,cC~~~,A~V~;~~(Mackey) with gVin V# V for any LEG. Thus ezE9’(A). Then, if XEA” and xY, c Y,, one has eixei =O, and x= 1x1 = (e,+e,)x(e,+e,) =e,x+e,xe,~.~~(A)x+e,x9~(A)~9”(A). Assertion (iv) is now proved. For the last point we have Y(V) = Y,(V) + Y,(V) with Y,(V) = 0 (assertion(i))and.f”(Y,)=Osince Y,Y=Y,;thus Y,(V)=Y,and Y(V)=Y,. Remark 1. In the above proposition, (i) and (ii), regarded as conditions on Y, and Y,, determine the type of kG-isomorphism of Y, and Y,: Y, z Y(V), and Y2 is a supplement of Y,. But it is easy to see that when neither is zero, they are not uniquely determined as subsets of Y. Though YF is canonic, Y:’ = 9 “( Y).

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MARC CABANES

A.2.3. The Algebra (End, Y)(Q). THEOREM

1

(Puig).

Zf Y is a p-permutation module and Q a p-subgroup (End, Y)(Q) and End,( Y(Q)) are isomorphic.

of G, the -Vo(Q)-algebras

Proof: We must point out an M,(Q)-algebra morphism from (End, Y)Q = End,Q Y to End,( Y(Q)), which is onto and of kernel 9Q(End, Y). Let us take the morphism induced by the natural action of (End, Y)Q on Y(Q) ( one knows that YQ and yQ( Y) are stable under every element of (End, Y)Q; this is an J&(Q)-algebra morphism). If one denotes res_,,cQ, Y= Y, @ Y, as in Proposition 2 for Q as a normal subgroup of J&(Q), then Y(Q) = Y, and it is clear that the morphism from (End, Y)Q to End,( Y(Q)) considered above corresponds to the map from End,, Y to End, Y, which sends x to e,xei (identifying End, Y, with {x’EEnd, Kx’Y,=O and X’YC Y,}). Its kernel is {xE(Endk Y)“; xY, c Y,}; from Proposition l(iv), this kernel equals @(End, Y). It remains to check that our morphism is surjective. Let y E End, Y, = Endko Y, (Proposition l(ii)). Then, one makes an element y’ of End,, Y by extending y to Y with 0 on Y,. It is quite clear that the image of y’ is y. Remark 2. We will denote by Brh the NG(Q)-algebra morphism from EndkQ Y onto (End, Y(Q)) considered above. Y = Y, @ Y2 as in Proposition 2 with Y(Q) = Y,, If one writes res _wcCQJ then, if x E EndkQ Y, Brb(x) is the projection onto Y, of the restriction of x to Y,. Let i be a primitive idempotent of End,,,,Q, Y contained either in e, or in e2. If it is contained in e, then Brb(i) . Y(Q) = i. Y, = iY, which is Q-trivial so, in this case, Brb(i) . Y(Q) = (iY)(Q). If i is contained in e2 then Brb(i) = 0 since iY c Y,, and by the definition of Brb above, we have also iY(Q)I Y,(Q)=O. Hence, in the two cases, Brb(i). Y(Q)((iY)(Q) as kJG(Q)-module. This is also true for any idempotent conjugate of i by an element of Autk_ri,CQ)Y, so it is true for any primitive idempotent of Endk.,cco, Y. By linearity, it is true for any idempotent of End,_,b,Q, Y that if i is any idempotent of Endk_,tiCQ)Y the following isomorphism of kJI/^,(Q)-modules holds:

Brb(i). Y(Q) N iY(Q). Remark 3. If Y = XOG Z, the restriction of the Brb associated with Y to the subalgebra (End,, X) x (EndkQ 2) of End,, Y is the direct sum of the Brys associated with X and 2 (take res_,i;cQ,X=X, OX, and res_,k_Q)Z = Z, 0 Z2 . . . ),

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BRAUER MORPHISM

A.3. AN EXAMPLE Let Y = indz 1 = kG OkH k for H a subgroup of G. Let Q be a p-subgroup of G. We study Y(Q) and Brb when p = char(k). A.3.1. Let .F(Q, H)=: {gEG; QRcHj. Then resk Y=

0 gtT(Q.

HI

4gOl)O

0

4gOl).

KC~lQ.Hl

One checks easily that each of the two terms is stable under J;(Q), and that the first is isomorphic, as TV;-module, to @.tcCQ,gHcfl,Q “) ind-'l(Q) .,cCQ)nx,,1, while the second is isomorphic to @,tcCQ,gHti .SCd,HJ ind,SI +,k(Q) GCQ,nnH 1 (we have, in fact, applied Mackey’s formula to res,,b,Q,indC, 1). Let us denote by Y, the first sum and by Y, the second. One has Y,(Q)= Y, and Y,(Q)=0 by [3, (1.3)] (Q acts trivially on Y,, while resQ Y2 can be developed easily by Mackey’s formula in a sum of transitive permutation Q-modules, which are all non-trivial since QngH d Q when g$S(Q, H).) Hence Y, and Y, are as denoted in Proposition 2. A.3.2. We give an explicit form of Brh on a subalgebra of End,o Y, namely End,, Y, by using a k-basis. Let us set a notation: DEFINITION 2. If G’ is a finite group, H’ a subgroup, and n an element of G’, let a, be the element of End,,. (ind$ k) such that a,( 10 1) equals Y{g@l;gEH’nH’}.

One knows that a,, = a,, if, and only if, H’nH’ = H’n’H’, and (a,; n E G} is a k-basis of End,,. (ind$ k). A.3.3. If n E G and a, E End, Y defined as above, Brb(a,) is the projection on Y, of the restriction of a, to Y,, where Y, = egerCQ, H) k( g@ 1) (see A.2.3, Remark 2). One checks immediately that Brb(a,) =0 when n $ S(Q, H), since then a, Y, c Yz. It is not difficult to give Brb(a,) when n E F( Q, H); we do that calculation in a particular case: PROPOSITION 2. Let G be a finite group, H a subgroup, and Q a p-subgroup of G such that H c S(Q, H) = &&(Q). Let Y = indg k for k an algebraically closed field of characteristic p. Then Y(Q) = ind$G(Q)k, and Brb sends a, to a, (as element of EndbtcCQ)(ind?(Q) k)) when n E J&Q), and to 0 if n g -4$(Q).

We have (see A.3.1) Y, =indzb(Q) k and Y,= @ga-+c,QJ Ifg@lEY1 withgE&(Q), one hasa,(g@l)=g(a,(l@l))= g.9’{g’@ 1; g’E HnH}. Proof.

k(g@l).

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MARC CABANES

So, when n+.&(Q), a,( Y,)c Y,, while, if neJv;,(Q), a,( Y,)c Y, and the expression above of the action of a, on Y1 gives well the a, defined in the permutation module ind$(Q) k in Definition 2. A.4.

THE GREEN

CORRESPONDENCE

P-PERMUTATION

IN THE CASE OF

MODULES

The following theorem gives precise information about the vertex of a given p-permutation module. Further results are in [3], among them, the connection with Green correspondence. THEOREM 2. Let Y be a p-permutation kG-module, Z be an indecomposable direct summand of Y, and V be a p-subgroup of G such that the kAJo(V)-module Z(V) admits a projective direct summand 71.Then:

(i) (ii)

n = Z( V) and it is indecomposable; VE vtx( Y).

Moreover, if one denotes by S, the irreducible representation of End,WvGC.,( Y( V)) associated with the direct summand 71of Y(V), one has: (iii) S, 0Br’,, gives an irreducible representation of Endkc Y, which is the one associated with Z. Proof Let A =: Endk Z. Since Z is a p-permutation module, A(V) N End,(Z( V)) as MG( V)-algebras; let Br’,, denote the .MG(V)morphism from A’ to End,(Z( V)) constructed in A.2. One has Br’JAG,) = (End,(Z( V)),p’?c”‘)since Br.(AG,) = (A( V))i”“). The algebra AG is a local algebra since Z is indecomposable, and A”y is a two-sided ideal, hence is nilpotent or equal to AG. If it were nilpotent, Br;(AG,) should be nilpotent, but Br’JAG,) = (End,(Z( V))fc(” contains the idempotent of End,,-,, ,,) (Z( V)) associated with the direct summand 71of Z(V) (this idempotent is in the relative trace ideal since n is projective: Higman’s criterion). Eventually A”, = AG. One gets also that Br’JAG) is a local subalgebra (image of a local algebra) in End,,,Cv) (Z(V)); this subalgebra clearly contains 1 and the idempotent associated to n. These idempotents must be equal, hence Z(V) = z. Point (i) is now proved. Point (ii) is also a straightforward consequence: VC vtx(Z) since Z(V) #O (see [3, 3.2(l)] and a vertex is in V since A$= AC (for definition of the vertex, see our “background”). (iii) S, 0Br’,, is an irreducible representation of EndkG Y since it is an irreducible representation of End,, Z = A G: S, 0Br’,, (End,, Y) 3 S,oBr’,, (AG)~SnoBr; (A$)=S,((End, ~)~(Y))=S,(End-~.~-,(.,~)=Im S,.

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BRAUER MORPHISM

Moreover, if iZ is the primitive idempotent of End,, Y associated with Z, and i, the one in End.,(;(,,, (Y(Y)) associated with n, one has S, 0Br’,, (iZ) # 0 since Br’,,(i,) is a conjugate of i, : Br; (iZ) . Y( V) = (iZY)( V) = Z(V) = rc.Eventually, S, 0Br’,, = S, since the first representation is irreducible and does not kill i,.

AS. CASE OF RANK ONE CHEVALLEY GROUPS We conclude the recapitulation of our needs in modular representation theory by a proposition concerning Y=: indG, 1, where U is a Sylow p-subgroup of a Chevalley group of rank 1. We have not yet defined these groups, which is why the hypotheses of the proposition may appear somewhat artificial. The result itself is not new and may be seen as a consequence of Green-Sawada theory. We prove it by elementary means. PROPOSITION 3.

Let G be a finite group with two subgroups T and U

satisfying:

(Hl ) (H2) (H3) (H4) (H5)

U is a Sylow p-subgroup (for a prime p) in G; T is an abelian p’-group normalizing

V;

denoting B= UT, one has IB\G/B( =2; -4’&(T) & B; G is generated by its p-elements.

Then, if k is an algebraically closed field of characteristic p, indg k is of width ITI + 1. Proof: Let n be an element of -F;(T)\B (by (H4)); then G= Bu BnB (by (H3)). By (H2), one has indB, l@/.tXk(T) 1, hence indz 1 = @zex,(r) indg 1. Moreover, if A E X,(T), the following holds: resBindg I= I @indi,,, (n . A) (Mackey’s formula), though these two kB-modules are indecomposable: their restrictions to U are 1 and ind:, nB(1 ), and [3,0.3] applies. So, if indg I is not indecomposable, it is the sum of exactly two indecomposable kG-submodules: indg ,I = M, @M, with ress M, = 1, and ress M, = indBBn”B(n .,I). Then MI is one-dimensional, though G is generated by its p-elements, hence M, = 1 and I = 1. We conclude that if 1 E X,( T)\l, ind,GA is indecomposable; and ind,G1 is of width inferior or equal to 2. This can be made precise: since p 1 (G : B) (Hl ), 11indg 1, so indg 1 is not indecomposable. Eventually the total width equals 2+ IX,(T)\11 = ITI + 1.

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MARC CABANES

PART B. MODULAR REPRESENTATIONS OF CHEVALLEY GROUPS The first way of classifying the irreducible representations of finite Chevalley groups in natural characteristic was to find them as restrictions of irredudible algebraic representations of the reductive group G of which G is the fixed point subgroup under an algebraic automorphism (see [27]). Another method was developed by Curtis and Richen (see [9]]), its general framework being the finite split BN-pairs. Curtis’ results involve certain expressions YX, . n, acting on M” (with M an irreducible kG-module and U the Sylow p-subgroup in the split BN-pair) and make it possible to index the irreducible modules by “weights.” The computations of Curtis’ paper were enlightened for the first time by Sawada [25], who showed that irreducible kG-modules are indexed by irreducible representations of a certain endomorphism ring. The action of the terms yX,n, on MU may also be interpreted as the action of certain generators of that ring which is the endomorphism ring of the permutation module ind5 k. Then Green [17] pointed out what was general in Sawada’s work: it is a remarkable result that for any finite group and under the hypothesis that End,,(indE k) is a quasi-Frobenius ring, the essential part of Sawada’s approach remains valid. The application to split BN-pairs of this setting is exposed by Tinberg in [28]. We intend to explain this whole set of theories and results while showing how Brauer morphism links the constructions for the group and those for the Levi subgroups; our methods make the simplification of certain points (existence of weights, Part C, etc.) possible.

B.6. GREEN-SAWADA THEORY B.6.1. Let k be an algebraically closed field, G a finite group, and Y a kc-module, and let Z = End,, Y. Let 1 = Cic, i be a decomposition of the unity of Z. Then there is a bijection between the conjugacy classesof elements in I and the irreducible Z-modules, the irreducible Z-module associated to i being the head of Xi. The conjugacy classesof elements of I are also the isomorphism types of indecomposable kG-summands of Y. Under certain hypotheses, Green showed that these summands iY of Y are, like the xl’s of irreducible head, with hd iY N Ghd jY if, and only if, i and j are conjugates.

BRAUER MORPHISM

11

B.6.2. The hypotheses on Y and X are the following: (i) (ii) (possibly (iii)

Y-, Y*, sot Y and hd Y involve the same irreducible kG-modules with different multiplicities), 2 is quasi-Frobenius.

The third hypothesis needs some explanation. In [22], Nakayama defines quasi-Frobenius algebras as k-algebras X such that, if 1 = xi,, i is a decomposition of the unity of 2, one has: VieZ, 3j, ~1 such that -yi”j, N (ix)* as %-modules. In fact he exposes it under the form of a permutation on conjugacy classesof elements of I: if i E Z one denotes by i the “conjugacy class” (i’ E I; 3.xE %‘“,x invertible and xi’x -’ = i}; there exists a permutation c on the set of conjugacy classesof elements of Z, such that: if ie I and je a(i) then Xi N (j8)* as X-modules. The following flows immediately: sot Xj E (hd i#)* and is irreducible, and: if j, j’ are in Z, ~j2:~j’Oj=jOhd~jhhd~j’ 0 a(j) = a(]) 0 sot Xj N sot Xj’. B.6.3. Keeping the decomposition 1 = Cie, i, Y= @,,,iY is a decomposition of Y in a direct sum of indecomposable kG-modules. One knows that: (iii’)

ix is a module-X with irreducible head and socle.

And if i, je 1 one has that i and j are conjugate o iY 2: jYo ix N j.X o hd i% ‘v hd j% o sot iAf N sot j%“. So one can see that the quasi-Fobenius hypothesis causes a sort of symmetry between head and socle, and also between iY and ix. Green’s theory will show how far that symmetry may go. The notations and hypotheses are as introduced in B.6.1. and B.6.2. According to Green one denotes by F(M), if M is any. kG-module, the module-z equal to Horn&Y, M). One has F(iY) = iX’ for any idempotent i in X’. Green’s results are the following (see [ 17, Theorems 1 and 21): THEOREM

3 (Green). Zf iE Z, one has:

( 1) sot iY and hd iY are irreducible; (2) F(soc iY) = sot iH and F(hd iY) zx hd ix. THEOREM 4 (Green). F induces a bijection between the irreducible kG-modules involved in hd Y and the irreducible modules-x.

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MARCCABANES

Remark 4. If itz I, the cardinality of i, “multiplicity of i in 1,” is the number of jYs (for j in I) isomorphic to iY. It equals the k-dimension of hd M, the associated simple representation of Z.

B.7. THE MODULAR HECKE ALGEBRA B.7.1. The Groups 7.1.1. For the remainder of this paper, p is a prime and G is a finite group with a split BN-pair of characteristic p. The hypothesis on G consists in the existence of three subgroups U, N, and T such that (see [24]): (i) U is a p-group and T is an abelian p’-group normalizing U; one denotes by B the semi-direct product UT. (ii)

T = B n N and (B, N) defines a BN-pair, i.e. (see [2]):

(ii’) Ta N and W= N/T is a Coxeter group (see [Z]); let R be its distinguished set of generators, (ii”) (ii”‘) (iii)

if r E R and USEW then rBw c BwB u BrwB while rBr # B, (B, N) = G.

flntN B”= T.

The order of R is called the rank of G. We denote by I the length map from WtoN,ifnENoneputsE=:nTEWandI(n)=I(E). The results we use in the sequel are among the elementary ones of [24,9], except for the Levi decomposition (see [lo] and also [30]). First there exists a root system @ (possibly non-crystallographic) with basis A such that W is the associated Weyl group; if c1E A one denotes by rr the corresponding element of R. One denotes by X, the subgroup U n Uworz of U, where w0 is the element of W of maximal length. Then X, # 1, and one denotes its order by qol, which is a non-trivial power of p, and sets q& =: ( U: X,). If u E A and w E W, the subgroup wX, w ~ i depends only on w(a) = 8, which is denoted by X,. If Y is any subset of @, we denote by X, the subgroup X,=: (X,; PE Y). One sets UP = X0-. One has the following properties: Pl. UnnUn-L c UnmUm-’ when n, m are elements of N such that -‘m); U is a Sylow p-subgroup of G. I(n)=l(m)+l(m

13

BRAUER MORPHISM

P2. If nEN, U=(UnnUn~‘).(Un~w,Uw,n~‘) associated decomposition of every element of U.

with unicity of the

P3. If c1E A there exists ncrE N n X, .X-, .X, such that fi, = rI, and we assume n, has been chosen so through the whole sequel. P4. If ZC R (or A as well), the parabolic subgroup P, is equal to the semi-direct product U, >aL,, where U,=: X,+,,: and L,= T. X,,. One has U,= O,(P,), P,= JQ U,) and L, is a split BN-pair of characteristic p and rank 111for the subgroups X,: , T. (n, ; c1E Z) := N, and T. P5. If Zc A, then G,=: (X,, CIE@,) is a group with a split BN-pair of characteristic p and rank 111 for the subgroups X@:, N,n G, and T1=: Tn G,. If Z= {M} one denotes TX=: TiZi = Tn (X,, X-,); since norE G, and T normalizes G, one has [n,, T] c T,. If Z= A one denotes G, =: G, = (X,; a E @), which equals also (U, U- ) since Vu, n, E ( U, Up ). One denotes T,, = Tn G,. P6.

N is a representative system of U\G/U.

B.7.2. x,(G,

U)

7.2.1. Let G be a finite group with a split BN-pair of characteristic p. Let A be a ring. One denotes by A the trivial AC-module, let Y, = A and yrO,(G, U) = End,, Y, (possibly abbreviated & or ind$A =AGO,, X). It is clear that xA(G, U) is A-free with basis (a,; n E G} (notation of A.3); one recalls that a,(g@l)=g~a,(l@l)=Y{g’@1;g’~gUnU}. One has 2’ = zr 0, A. From P6 it suffices to express the products anam for n, m E N to get the complete law of SA, and if n, m E N a,, = a,,, on = m. 7.2.2. The following theorem is a slightly different formulation theorem of Sawada [25,2.6)]:

of a

THEOREM 5. For every c1E A there exists a map z,: T, + N such that ,EToz2(t)=qm1 and: c

(i) anam = a,,,” if n, m E N with l(mn) = l(m) + I(n), (ii) a,~,~= qzan,.n+ c,, r, z,(t) am when n E N and l(n, .n) < l(n). As a consequence (a,; t E T> is isomorphic to AT and ,& is generated bJ the set (a,; t E T} u {a,,; QE A}.

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MARC CABANES

Proof: Omitted. It follows the same lines as in [25]. The key arguments for (i) are Pl and P2. Part (ii) is proved by reduction to the case G = G, = (X,, X-,), using P5 and defining z,(t) as the cardinality of n,X,n, A X,tnrXa. Remark 5. In the general case of two elements n and m in N, an easy induction on f(m) leads to the fact that anam is a linear combination of those a,,, such that Z(n’) < I(n) + Z(m) or n’ = mn. Remark 6. In Zo(G, U), let e = I TI -’ C,, Ta,; then e is an idempotent and it is straightforward to check that e . indG, 1 is a direct summand of indG, 1, isomorphic to indg 1 (see also our Remark 9 in B.9.4.). Then EndoJindg 1) is isomorphic to e#o(G, U) e. The latter algebra with unit e is also Zo(G, U) . e since e is clearly central (compute a,. e by means of Theorem 5(i)). The surjection of Q-algebras defined by

ZdG, U) + XQ(G, U) . e

has as kernel the ideal Xo(G, U) . (a, - Id; t E T). So, the classical Hecke algebra Z&(G, B) is isomorphic to Zo(G, U)/&o(G, U) . (a, - a, ; t E T); in this quotient the class of a,, depends only on w = nT, and if one denotes by a, this class, Theorem 5, gives in %o(G, B):

0)

a,a,., = aw,,w,when I( w’w) = 1(w) + 1(WI’),and

(ii)

a,,,a,E= qrargw,+ (4%- 1) a, when tl E d and l(r,w) < I(w).

These formulas are the classical ones about Xo(G, B) (see [ 12, 211). 7.2.3. If A is a field of characteristic p, the multiplicative law of sA(G, U) gets simpler than indicated in Theorem 5. Since then qZ . 1A = 0 for every u E A, one has azu=C,,,Z z,(t) a,,,, for z,(t) E Z. 1, and c ,cTz,(r)= -15 while a,, a,,, = am,, whenever n, me N with f(mn) = Z(m) + Z(n). A consequence is that if f(n,n) < l(n), a,anb = acna,-l ,JaJ* = c tsTzz,(tb,, = anCLcT, z,(t)a,), whileifI(nn,) < Z(n), aBaa, = (a,J2a,+,,-l = zh( t) a,a,, where (C tt T, z,(f) a,J a+,)“x:rtT ‘htt) afanaan(n&l=CtcTm ~h(t)=z,(n,t(n,)~‘)(n,T,(n,)~‘= T,) with C,, Tnz,(t) =CltTm z&(t)= - 1. From this elementary material let us show the existence of a p-block of defect 0 for G (the classical proof of this result which goes back to Steinberg seemsto be essentially different). Let A = k an algebraically closed field of characteristic p, let x be an element of X,(T) = Mor(T, k*) such that x(T,) = 1 for every TVin A, i.e.,

15

BRAUERMORPHISM

such that x(T,)= 1 if T,=: (T,;cr~d). Let e,=: ITI-‘C,,.X(~-‘)~,; then e, is clearly an idempotent such that VIE T, atex=e,a,=X(z) e2. Let r a, ... ra, be a reduced expression of w,, in W and let us define n, =: n,, . . . n,,, which is an element of maximal length in N. Let us define ek =: (- l)lcno)e,a,,e,; then it is easy to check the following: LEMMA. If r~Tanda~A, a,ei=e;a,=x(t)el, Then e’ is a central idempotent in &(G, U).

and a,rei=e;a,Z=

-ek.

We are going to show that the direct summand ek. Yk of Y, = indg k is an irreducible projective kG-module; this gives as many p-block of defect zero as characters x of T such that x( TR) = 1 since if x # x’ek. Y, S&Z ek,. Y: the contrary should imply that the primitive idempotents ei and ek, are conjugates, but they are central and distinct. We begin with the projectivity of ek. Y,; one has to check that ek is in the relative trace ideal (end, Y,)f in Xk. Let s0 be the element of end, Y, which sends 10 1 on n,@ 1 and each g@ 1 for ge G\U on 0; one has TrF(c,) = a,,, hence a,,,,E (End, Y,)p and therefore e;E (end, Y,)p since this relative trace subspace is a twosided ideal of $. Irreducibility of ek . Y, : if M is an irreducible submodule let us prove that M= ei. Yk. One has O# ML’= of e;.Y,, Mor,,(indz 1, M) (Frobenius reciprocity), hence there exists x E Xk(G, U) such that x Y, = M; thus ei x = x. But kej, is an ideal, hence ek .x Y, is 0 or ei. Y,; therefore M = ei Y, as announced. 7.2.4. A first consequence of Theorem 5 above is that for any ring A, &(G, U) is a Frobenius algebra, hence a quasi-Frobenius algebra. COROLLARY 1 (Green).

XA(G, U) is Frobenius.

Proof. Take the linear form giving the coordinate on a, and use Remark 5 (see also [28, 3.73).

7.2.5. If G is a finite group with a split BN-pair of characteristic p, from now on, let k be an algebraically closed field of characteristic p. We will denote Y =: Y,(G, U) = indG,k, X or ??(G, U) =: &(G, U) = Endk, Y. The first remark to be made is that X and Y satisfy all the hypotheses of B.6.2, (Green-Sawada theory). The isomorphism Y z Y* is clear. Condition (ii) flows from the fact that every irreducible kG-module M is in hd Y and in sot Y: Horn& Y, M) = Horn&k, resu M) = M” #O and Horn&M, Y) = Hom,,(resUM, k) = M/( 1 - U) M# 0. From Corollary 1 above, condition (iii) of B.6.2 is also satisfied. Then Theorems 3 and 4 are true in this context; one notes that the set of irreducible kc-modules involved is in fact the set of all irreducible

16

MARC CABANES

kG-modules. So Theorem 4 gives a parametrization of irreducible representations in natural characteristic by irreducible representations of the algebra &, where explicit computations are now possible. For instance, Remark 6 following the theorems mentioned above can be made precise: COROLLARY

2 (Sawada). Every irreducible right module for #

is of

dimension 1. Proof. Let M be an irreducible module-x. The subalgebra (a,; t E r> of %? is isomorphic to kT, which is commutative split and semi-simple. Then M is a direct sum of lines, each of them stable under (a,; t E T). Let D be one of them. Let n, be an element of maximal length in N such that a,, . D #O. Let D’= u,, . D. One proves easily that D’ is %-stable, hence equal to M (see [28, 3.133).

7.2.6. From this corollary we deduce that, in this context, if Z is a decomposition of the unity of X, every iY for ie Z is isomorphic to no other jY for je Z. So Theorems 3 and 4 take the following form: Let Irr(%‘) be the set of ail irreducible representations of 2”; then Y($) with every Y($) indecomposable and M($) =: Y=@ ILEITTCJlr) hd Y($) irreducible (kG-modules). One has $ = Horn,& Y, M(+)) and the correspondence Ic/-+ M(ll/) is l-l from Irr(2) to the set of all irreducible kc-modules. B.8. BRAUER MORPHISM BETWEEN MODULAR HECKE ALGEBRAS Throughout this part, J is a subset of R. Then P, = J,(U,), U, = O,(P,); and L, = T. X,, has a split BN-pair for the subgroups UnL,=X,;, T and NJ = N n L, (see P4 in B.7.1). We now consider Sk(LJ, Un L,) ad point out a link with J’$(G, U). B.8.1. Let us compute Y( U,) (notation of A.l) as kP,module. The following is standard: LEMMA

1. y( UJ, U) = P, = A$( U,).

Then Y( U,) is easily computed (see A.3): it is ind$ 1. This Prmodule is trivial under the action of U,, so it is Y,(L,, Un L,) = indtin L, 1 as L,module. B.8.2. Following the method of Part A.3, the Brauer morphism can be made explicit:

BRAUER MORPHISM

17

THEOREM 6. Zf J c R, the Brauer morphism Br, induces an algebra morphism from zk(G, U) onto Xj(LJ, X,7), denoted by Br, too, such that ifneN,

Brci,(an) = a, =o

ifnEN, if not.

Proof: We recall that, in A.l-A.3, when Y is any p-permutation kG-module and Q any p-subgroup of G, one has an isomorphism of ,6’;,(Q)-algebras between (End, Y)(Q) and End,(Y(Q)), so the Brauer morphism Br, from End,, Y onto (End, Y)(Q) induces a morphism, denoted by Brb, from Endka Y onto End,( Y(Q)). We have computed in A.3, Proposition 2, the restriction of Brb to End, Y in the case when Y = indg 1. Theorem 6 above simply expressesthe result of Proposition 2. B.8.3. As a first application, the following corollary shows how to imbed Xk( L,, X@;) in Xk( G, U). 3. Zf JC R, the set {a,,; n E N,) generates a subalgebra of ,Yink(GU) equal to 0 nE,.,,k. a, and isomorphic to zk(LJ, X,7). COROLLARY

ProofI It is clear from Theorem 5 that on EN,k . a, is a subalgebra of %‘. The isomorphism with Xk(LJ, X,:) is given by Br,,, (see Theorem 6).

B.9. THE IRREDUCIBLE REPRESENTATIONS OF Hk(G, ci) B.9.1. In B.7.2.5 one has seen the interest of knowing the irreducible representations of Xjk(G, U). Such a representation of X? is known to be one-dimensional (Corollary 2) so it takes the form of a ring homomorphism I,$from sk to k. It suffices to know $ on the set of generators {at; t E T> u {a,=; c(E A}: The restriction of $ to {al; t E T} is an element of X,(T): x such that $(a,) =x(t). Once x is known, one has (I/(a,,)Ek with +(a,=)* = $(a:@)= (C ,ETzz,(t) x(t)) ti(a,J, hence $(a,J must equal zero or C,, Tsz,(t) x(t), the latter being clearly - 1 when x( r,) = 1. DEFINITION 3. Notation. (i) If x E X,( T) = Mor( T, k*) one denotes by R(X) the subset of R equal to (ra; x( T,) = 1}. (ii) If + is an irreducible representation of Xk, we call its “associated pair” the pair (x, S), where x E X,(T) is defined as in the above discussion, and S=: {rr; $(a,,,)#O}.

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MARC CABANES

(iii) A pair (x, S) belonging to the Cartesian product X,(T) x B(R) is said to be “admissible” if, and only if, SC R(X). (iv) If Ic/is an irreducible (one-dimensional) representation of Xk, $ is said to be “admissible” if, and only if, its associated pair is admissible. Remark 7. From the above discussion, the associated pair of a onedimensional representation of Xk determines it uniquely. In the case when this pair is admissible, +(a,J is either 0 or - 1, the value being - 1 if, and only if, ra belongs to S. The number of admissible pairs with second term S is clearly (T: T,) with the notation Ts = (T,; rorE S). So, the number of admissible pairs is c sc R (T: Ts).

B.9.2. We now check that for any admissible pair there exists a onedimensional representation of Xk whose associated pair is this one. That is the part of the theory which is called in [24] “existence of weights.” This existence was proved explicitly; our proof will be explicit for admissible pairs of type (x, R), the Brauer morphism allowing one to deduce the general case without further pain. The converse comes easily from what we have done about the rank one case (A.5) and gives some extra information about the structure constants z,(t)‘s. THEOREM 7 (Sawada). For every admissible (x, S) there exists a morphism $(x, S) f rom X? to k (i.e., a one-dimensional representation of &) whose associated pair is (x, S). Conversely:

(a) (b)

every irreducible representation of Xk is admissible, ifcc~d

and tE T,, z,(t)=

- (T,I-’

(mod p).

Proof: (i) First assertion in the case S = R. Since (x, R) is admissible, one has x( T,) = 1 for every c( in d; then one defines ek as in 7.2.3 and Lemma 2 in the same section applies, whence kek is clearly a one-dimensional representation of Xk, as a left ideal, with associated pair (x, R). (ii) First assertion in the general case. Applying (i) above to the group L,, we get a k-algebra morphism I,+&, S) from &(Ls, X,;) to k such that: for any t in T and rr in S, Ic/&, S)(a,) =x(t) and t+Qs(x,S)(ann) = - 1. Let $(x, S) =: Gs(x, S)o Brvs, which is a morphism of k-algebras from Yk(G, U) to k. From the explicit value of BrOs (see Theorem 6) on the generators of J$(G, U) it is clear that I&, S)(a,) = x(t) if t E T, $(x, S)(aJ= - 1 if raES, and $(x, S)(a,=) =0 if r,E R\S. The associated pair of $(x, S) is (1, S). Now we know that every irreducible admissible representation exists; proving assertion (a) consists in checking that no other exists.

BRAUER

MORPHISM

19

(iii) Assertion (a) for IRI = 1. The number of summands of a decomposition of Y is bigger or equal to the number of irreducible representations of End,, Y (bijection between the conjugacy classes of primitive idempotent and the irreducible representations of the algebra Endk, Y). If yi”k has more irreducible representations than the 1TI + (T: 7’,) admissible ones (see Remark 9), the width of Y is then strictly greater than (T( +(T:T,).

On the other hand, let us apply Proposition 4 of A.5 to the group G, = X,,, -Si = (U, Up ), which satisfies the hypotheses: one yields wth(ind3 1) = (T,J + 1. But resG, Y = resG,indG, 1 = eGSgL,indG; 1 = and T,=TnG,. So wth(Y)< (T: T,) ind2 1 since G=T.G, wth(res,* Y) = (T: Ta). wth(ind2 1) = 1TJ + (T: T,), a contradiction. (iv) Assertion (a) for arbitrary rank. Let $ be an irreducible representation of Xi. From Corollary 2 it is a k-algebra morphism from Sk to k. Let (x, S) be its associated pair; we have to check that S c R(X). Let r2 E S; then one has to prove that x( T,) = 1. By Corollary 3 in B.8.3 the subalgebra @ nE,,,,k .a, of %$ is isomorphic to &(L=, X,) by “a,, t+ a,,.” The restriction of II/ to this subalgebra is an irreducible representation of Xj(L*, X,); by the rank one case it is admissible, hence I( T,) = 1 since tj(a,J # 0. The proof of (a) is now complete. (v) Assertion (b) is not essential to the aim of this paper, so we present only a rapid sketch. One has first z,(tt’) = z,(t) when t E T and t’ E [T, n,]: by writing a,(aJ* = (~2,~)’a,v (since ~,a~~ = an,dba = a,zua,T,rH-I~u,T = an,an2sn-l = a,* a.snx= u&a,) and making explicit the coefficient ‘of a,,,,, one obiains ~,(t)=z,(tsn,s~‘n;~) for any t in T, and any s in T. Then in s$: aiz = u,,~x = XU,~ for x = C,, TUz,(l) a,. On the other hand, if t’ E T, the element ai* + a,,,,, is clearly killed by every irreducible representation $(x, S) of Xk so it belongs to the Jacobson radical. For instance, a:, + aRa= anU(x+ 1) must be nilpotent: there exists m in N such that (u,~(x + 1))” = 0, i.e., u,%x”- ‘(x + 1)” = 0, hence x2 +x is nilpotent in the commutative semi-simple algebra @,, T ku,, so x2 = -x. One has also ai%+ ( I T,I - ’ C, 6 r, a,) a,& in the Jacobson radical, and, for the same reasons as above, x2 = - 1T,I -I C,, Tza, . x. But the latter expression is IT,Ip’Cter,a, since CrcTaz,(f)= -1. So x= -ITa,lplC,,T,ar. The proof of Theorem 7 is now complete. When (x, S) is an admissible pair for G, we denote by $(x, S) the associated admissible one-dimensional representation of Xk; we recall that t&x, S)(a,) = x(t) if I E T, t&x, S)(a,J = - 1 if rz E S, 0 if not. If $ = (l/(x, S) one denotes Y($)= Y(x, S) (see 7.2.5). One knows that hd Y($) is irreducible; we denote it by M(x, S). One has $(I, S) = Mor,,( Y, M(x, S)) as module-Xk.

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CABANES

B.9.4. Remark 8. We have seen in the proof of the above theorem that z, depends only on the classesof T,/Tz, where Tz = [T, n,]. In the case when L, is of type GL,, this implies that =a is constant: in GL,(p”) the commutator of the torus T with n, = (7 A) is ( ( j.ib ’ j.!‘,,r);A, p E lF$}, which is the whole T, = Tn SL,(p”). In the case when p = 2 and L, is of Suzuki type, Tz = TX since ra acts by inversion on T, [ 14, p. 181, which is a 2’-group. Hence Z, is constant. In the general case, the integers z,(t) for t in T, can be investigated by computing C I t rmA(t- ‘) z,(t) for every A E A’,( T,); this has been performed by Howlett and Kilmoyer in [lS, Sect.41; their result shows that when L, is of Lie type different from GL, or &(2*), there exists 1 in X,(T,)\l such that c,, TzA(t- ‘) Z,(Z) # 0, hence, in these cases,‘7%is not constant. Remark 9. Looking at the decomposition indz 1 = Y= @ Y(x, S), the sum being taken over the set of admissible pairs (x, S), two questions arise naturally. Since (G: U) is prime to p, the trivial kc-module 1 is a direct summand of indf, 1. What is the corresponding admissible pair? On the other hand, since B= U X T with kT split, commutative and semi-simple, one has indf, 1 = @j.EXICT)A. How does the decomposition @ Y(x, S) rearrange to give 0). indg A?

We give in 310the idempotent e inducing (resp.) eY = 1 and indg 2; then, the direct summand eY is isomorphic to etiCx, SJ(r)f,, Y(x, S) by the definition of Y(x, S). In the case of 1, if r is a representative system of G/U, one has indE 1 =k(C,.,gO1)OG{~gE~.~O1; CgtrAg=O} with the first summand isomorphic to 1. The projection of 10 1 is (G: U) ~ ’ C, Er g @ 1 = (G:U))“+?{g@l; geG}, which equals by definition of a,: (G:U)-’ so the projector equals (G:U)-‘CHENa, (they are c nENa,(l@l), G-morphisms and agree on 1 Q 1); one denotes it by e,. One has I+Q1, fa)(e,) = (G: 17-i 1TJ # 0, so, since e, is primitive, 1 = Y( 1, fa). In the case of ind,G1, one has kBQ,, k = @lEXktB, k(C,, .A(t-‘) t@ 1) with ~‘.x~=A(~‘)x~. when ~‘ET and ~~=:~,~~A(f~‘)f@l. One has SO kG@,,k= @ikG..Yi. with kG.x,= 101=IT’C,.,,,,,x,~ kG Oke k.u, N indg A. The projection of 10 1 on kG . xi. is (T( -’ .x2, which in turn equals 1TI ~ ’ C,, T A(r ~ ’ ) a,( 10 1). So the corresponding projector equals e,=: ITIplCIETE,(tpl)al. If (x, S) is any admissible pair, one has $(x, S)(e,)= ITl-lC,E.A(tpl)~(t), which is non-zero if, and only if, il=x. So indg il = @ s. R(A,Y(1, S).

21

BRAUER MORPHISM

B.lO. THE IRREDUCIBLE REPRESENTATIONS OF G Let M= M(x, S) be an irreducible representation of G; all the information on M is contained in the fact that Horn&Y, M) is an irreducible representation of 2 which is one-dimensional and equals $(x, S) with: \cl(x, S)(a,) = x(t) for any t E T and, if r,~ R, $(x, S)(a,J = - 1 if r,E S, cl/(x, S)(a,J = 0 if rr E R\S. In order to yield information on M itself, one has to see how 2, in fact its generators, acts on Hom,,( Y, M) = F(M). B.lO.1. Let us recall that F(M)=Hom,,(indg Horn& 1, res, M) = M”.

1, M) is identified with

PROPOSITION 4. Through the above identification, the action of Z@ on F(M) by composition on the left-hand side becomes the following action on M”: ifn~Gand.u~M”, a,,.x=Trg,.,,(n.x).

between Hom,,W Oku k M) Proof The identification and Hom,,,(k, res, M) (“Frobenius reciprocity”) consists in sending f E Hom,,(kGOkU k, M) on f( 10 1) (as image of 1, or viewing directly Hom,,(k, resu M) as Mu). If nEG, a, acts on F(M)=Hom,,(kG@,,k, M) so the proposition consists in checking that if by a,.f=foa,, f~Hom,,(kGO,,k,M),foa,(lOl)=Tr::,.,,(n.f(lOl)). Since f is a G-morphism TrE,.,,(n.f(l @l))=f(TrE,.Jn@l)) but Tr”On”Jn @ 1) is nothing but a,,( 10 1) :a,( 1 @ 1) is by definition (A.3.2, Definition2 cp{un@l; UE U} with un@l =u’n@l if, and only if, u‘-‘u E U n “0: This finishes the proof of the proposition. B.10.2. The following theorem enumerates the major properties of the irreducible kG-modules, all of them being consequences of GreenSawada theory, classification of irreducible representations of 2, and Proposition 9. THEOREM 8 (Curtis and Richen). Let G be a finite group with a split BN-pair U, N, T of characteristic p, let k be an algebraically closed fieId of characteristic p, and let M be an irreducible kG-module. Then:

(i) (ii) YX,.n,.m=O

(iii)

M” is a line km.

If T acts on MU by x E X,(T)

and if

rrr E R, one has

or -m, and it is zero ifr,$R(x). M=kU-

.m.

Proof: Assertion (i) comes from Green-Sawada theory: M” = F(M) so it is an irreducible representation of 2 (see B.7.2.5); hence it is one-dimensional (Corollary 2 of B.7.2.4). (ii) See a generalization below (Proposition 4). (iii) Since M is irreducible, since G is generated by U- . T

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MARC CABANES

and the n,‘s, it suffices to show that kU . M” is stable under the action of every n,. Let LIEA. First we check that if XEX, then n,x.ML’CkUMU. If x#l, thenxEL,\TX,soxEX,n,TX,andn,xEX~,TX,cU.B, hence n,x.M”cUB.MUckU-.M”. Case x=l:kXp,.MU is a non-zero (X-, is a p-group). One has kX-E-module, so (kX-, .MU)X-z#O kX_, . MUc MUa since MUc Mu= and X_, normalizes U,. Then (kX_, . M”)x-z c Muzxma = nor. MU since ““U = U, . XP,. But nor. M” is a line, so we have an equality instead of the latest inclusion; therefore n, . MU c kX, . Mu c kUMU, which completes the first step: Vx E X--,, n,.x.MUckUThen kUM”

.MU.

is stable under the action of n,: if u- E II--, one with XEX-, and ~EU-n(U)“‘; then n,u-*MU= n,xML’, which is included in kU-M” since n,xM” c kU- M” (first wn2 U. step) and n,yn;‘E Another proof of the first step is the original one due to Curtis and Richen (see [9, p. 211). has up=yx

B.10.3. The following proposition sharpens assertion (ii) of Theorem 8. A corollary is the characterization of defect 0 irreducible kG-modules. Another application is in [7, Sect. 51. DEFINITION 4. If n E N, let R, be the set of elements of R involved in a reduced decomposition of fi. PROPOSITION 5. Let G be a finite group with a split BN-pair of characteristic p, and let k be an algebraically closed field of characteristic p. Let M be an irreducible kc-module. If (x, S) is its associated admissible pair, one has: if no N and XE MU\O, then Trg,.Jn.x)=Y(Un”U).n.x, which equals 0 is, and only if; R, d S, while, if R, c S, Tr:, .,(n . x) = x( t)( - 1)““’ .xfor tETsuch that ntt’E(n,;r,ES).

Proof Ur\ “U is a representative system of U/Un “U, so Tr~,.t,(n.x)=Y(Un”U-)n.x. One has Trg,.,(n-x)=$(X,S)(a,).x by Proposition 9, with $(x, Wa,) = x(t). Ii/(x, S)(a,J :. . $(x, S)(a,a,) if n = n,, . . nor,.t with t E T and fi = ra, . . . ra, a reduced decomposition of fi. It is clear that $(x, S)(a,) =0 if, and only if, R, = {ral, .... r%,} ti S, and equals x( t)( - 1)’ when R, c S. COROLLARY 4 (Curtis and Richen). Let A4 be an irreducible kG-module associated with the admissible pair (x, S), then M is projective if, and only if, S=R. In this case Y(x, R) = M(x, R) and its dimension is 1UI.

BRAUER MORPHISM

23

ProojI Let M” = k. m. Since U- is a Sylow p-subgroup of G, M is projective if, and only if, res,- M = kU- *rn is projective. But resL/-M = equal to kU- .rn z kU-/I, where I is the left ideal of kU(aekU-;

am=O).

If X is a ring and I any left ideal, X/I is projective as X-module if, and only if, I = 0 (X -++ X/Z must split). Here kU- has a unique minimal left ideal: kY Up ; so M is projective if, and only if, YU- .rn # 0. But YU- .m=n,‘.Y(UnnoU-).n,.m (where n, is an element of N of maximal length) which is different from zero if, and only if, R = R,, c S (Proposition 10). Eventually, M(x, S) is projective if, and only if, S = R. In this case res,- M N kU-, so its k-dimension is (Up 1= 1UI. The equality Y(x, R) = M(x, R) flows from the fact that Y(x, R) is indecomposable with its head M(x, R) projective, hence a summand. As a matter of fact the problem of determining the irreducible projective kc-modules has been encountered in B.7.2.3, where we constructed for each x such that R(X) = R an idempotent el, such that el, Y is irreducible and projective. It is easy to check that if (A, S) is any admissible pair $(A, S)(ei) is 1 if, and only if, S= R and 2 =x; so eI, Y= Y(x, R), which tells us that in B.7.2.3. we had already checked that Y(x, R) is irreducible projective. B.ll. FURTHER RESULTS In what follows, let G be a finite group with a split BN-pair of characteristic p, and k an algebraically closed field of characteristic p. Let 2 = Zk( G, U); ifj c R, let YJ = ind 2, d = indc k. If (x, S) is an admissible pair with S c J, it is an admissible pair for the finite group with split BNpair L,; one denotes by Y,(x, S) the corresponding direct summand of Y,, M,(x, S) its head, eJ(x, S) the associated one-dimensional representation of Xk(LJ, Un L,). B.11.1. Tinberg’s Induction Formula In [29], N. Tinberg proved the formula indg, Y,(x, S) =

Y(x, S’). @ .s’L R(X)

JnS’=S

The proof consists in noting that ind:, Y,(x, S) is a direct summand of ind:, Y,= Y, so it is isomorphic to a sum @ Y(x, S’) for the pairs (x’, S’) ranging over a certain subset B of the set of admissible pairs for G. Then WndZ, Y,(x, S)) equals 0 (xsS’,EQM(x’, S’), and (x’, S’) E d if, and only if, Hom,(indF, Y,(x, S), M(i’, S’)) #O. By Frobenius reciprocity, the

24

MARC

CABANES

latter is equivalent to Hom,,( Y,(x, S), resp, M(x’, S’)) #O, which, in turn, is equivalent to hd Y,(x, S) 4 sot resp, M(x’, S’) since hd Y,(x, S) is irreducible: it equals M,(x, S). The socle of resp, M(x’, S’) is clearly kP, . (M(x’, S’) “) since M($, S’)U is a line. Moreover, (kP,.M(f, S’)“)U=M(~‘, S’)” and the associated pair for L, is (x’, S’ n J). So kP, . M(x’, S’)U = M,(x’, S’ n J), and (x’, S’) E 8 if, and only if, xf = x and s’ n J= S. B.11.2. A Dimension Formula Another problem is to determine the dimension of Y(x, S), and its p-adic valuation. Tinberg solved this problem by a formula (see [29]) shown below which relies on the fact that Y(x, (21) is one-dimensional when R(X) = R.

We present first another formula needing less material and leading directly to the p-adic valuation. If x E X,( T) and Jc R(X), by Tinberg’s induction formula, inGJ YJk J)= ORtX,3SzJY(x, S). But dim, Y,(x, J) is 1U n L,I by B.10.3, Corollary 4, applied to L,. Hence, if one denotes ds = dim, Y(x, S) for SEY(R(X)), one has for every JE.Y(R(x)) 1 d,= lUnL,I JCS

.(G:P,),

which inverts under the form

ds= =

1

t-1)

1

C-1) IJ”’ (G:P,)&I.

R(x)=JzS

IJ\sI (G: P,)(Un

L,(

R(x)IJzS

Eventually, if (x, S) is an admissible pair, the following holds: IX,; 1~’ dim, Y(x, S) =

1

(- l)‘J”s’ (G:P,)(X,;:X,;).

J:R(x)=IJJ~S

In particular IX@;1 divides dim, Y(x, S) and the quotient is equal to (G: P,) modulo p (in the summands for J $ S the index (X,; :X,;) power of p). Since P,3 MG( U), we get: is a non-trivial IX?;! ~ ’ dim, Y(x, S) = 1 mod p. In particular the highest power of p dividing dim, Y(x, S) is equal to lX,;l. B.11.3. From G to GO Denoting G, = ( U, U ~ ) (see B.7.1, (P5)), we are going to link up Y(G, U) (and its summands) to Y(G,, U) (and its summands).

25

BRAUER MORPHISM

If M is an irreducible kc-module, resGoM is still irreducible since MU is a line and M= kU- .M”= kG,,.MU (see B.10.3). If (x, S) is the admissible pair for G associated to M, then it is clear that (res, x, S) is the admissible pair for G, associated to resGoM (actions of r, and of the 9X,, n,‘s on MU): resGoM(x, S) = M(res,, x, S). Since G, u G, the Mackey formula gives res,, Y=res.,indEk=(G:G,).ind~nk=(Y,(G,,

U))‘G’Go’,

so wth(res,, Y) = (G:G,) . wth( Y(G,, U). But

hence wth(Y(G,, U))=&,.(T,:T,), and, since G,T=G and (G:G,)= (T:T,), wth(res,, Y) = (T:T,).&,.(T,:T,) = &,(T:T,) = wth(Y) (see B.9.1, Remark 9). Then, every indecomposable summand of Y remains indecomposable when restricted to G,. From the identification of res.,(hd Y(x, S)) = resGoM(x, S) seen above, we can recognize resGoY(& s): it is YG,(res,,,x, s). B. 11.4. One-Dimensional Modules Let (x, S) be an admissible pair for G; then Y(x, S) is of dimension 1 if, and only if, res@ Y(x, S) = 1 (G, = ( Ug; g E G ) ). But resGoY(x, S) equals Y,,(res,, x, S), which is the trivial module if, and only if, x(T,) = 1 and S = /zr (see B.9.4, Remark 11). So Y(x, S) is one-dimensional if, and only if, x( T,,) = 1 and S = 0. There are (T: To) such modules. The criterion is the same for the M(x, S)‘s: if x( T,) = 1 and S = 0, it is clear that, since Y(x, a) is a line, M(x, 4) is also a line. There are no more one-dimensional kG-modules; such a module would be clearly of vertex U and source 1, hence a summand of Y, hence among the Y(x, S)‘s. In fact, it is true that T, = T,. So Y(x, 0) is one-dimensional as soon as x(T,) = 1 for every crud, i.e., R(X)= R. Let x be an element of X,(T) such that R(X) = R, we are going to check that M(x, 4) is one-dimensional. Then it will be clear that 1 = resG,,M(x, 0) = M(res,, x, a), so I( T,) = 1. This holds for any x such that x( TR) = 1, hence T, = T,, since T, c T, is clear. We prove that if M = M(x, (ZI), M” is stable under G. This line is stable under B, hence it is enough to show that it is stable under any minimal parabolic, and that for any fundamental root ~1E d, M” is LX-stable. As a first step, kL, . M” is an irreducible kL,-module: (kL, . WQXa = M” since kL, . M” c ML’,. As kL,-module, kL, . M” is of associated pair (1, I$) since

26

MARC

CABANES

T acts by x on M” = (kL, . A4”)xu and .9X,. n, . M” = 0 since M= M(x, Qr). We have x( T,) = 1, hence kL, . M” is one-dimensional by the criterion of this section. Eventually kL, . Mu = M” as claimed.

B. 11.5. Tinberg’s Dimension Formula If (x, S) is an admissible pair for G, the following holds:

ind& Y,(x, 0) =

0 m s’) S’c R(x)\S

(applying Tinberg’s induction formula). So Cs:scsz d, = (G:P,(,,,,) since Y&, 121)is one-dimensional, x(Tn G,) = 1 since x(T,) = 1. A result of Solomon which extends easily to split &V-pairs expresses (G : PJ) for any parabolic P, as the sum C 2 (U:Un”U). JcK WH,(J)C@)+ w(R\J)

c @-

Comparing the two sums, one obtains dim, Y(x, S) =

c

(U: Un “‘U).

1: wfR(~)\S)c@+ w(R\R(~))un,(S)i@-

PART C. THE FAMILY OF THE IRREDUCIBLE PROJECTIVE REPRESENTATIONS OF THE LEVI SUBGROUPS The framework of the sequel is the one defined in Part B: G is a finite group with a split BN-pair of characteristic p and k is an algebraically closed field of characteristic p. C.12. THE GREEN CORRESPONDENT OF Y(x, S) C.12.1. Let (1, J) be an admissible pair for G: 1~ X,(T), Jc R(I). Then M,(,$ J) (notation of B.11) is an irreducible projective kCrmodule and it is a direct summand of Y, =: indL,J,LJk. One has Y, = Y( U,) (see B.&l ), so there exists an admissible pair for G, (x, S), such that M,(A, J) 1Y(x, S)( 17,). We may then apply Theorem 2 in A.4 for V= U,, Y = indg k, Z = Y(x, S) and rc= M,(n, J); one yields: (i) (ii) (iii)

Y(x, Wu,)

= MAA 4;

U, is a vertex of Y(x, S);

$A& 4 0Bru, = $(x, 9.

BRAUER MORPHISM

27

But tiJ(l, J) 0Br”, is easy to calculate: since Br,,(a,) = a,, if n E N, and 0 elsewhere (see B.8.2, Theorem 6), +,(A, J)o Br, is clearly cl/(& J). Hence the assertion (iii) implies: x = 1 and S = J. The following proposition now combines (i) and (ii): PROPOSITION 6. If (x, S) is any admissible pair for G, then U, is a vertex of Y(x, S) and Y(x, S)( U,) = M,(x, S) as kL,-module.

The vertex of Y(x, S) was computed first by Tinberg in [29]; as explained in the Introduction, our method is not essentially different from hers. C.12.2. Another application of Theorem 2 of Part A is to give an elementary approach (using only the Levi decomposition in a split BN-pair) of the following problem: PROPOSITION 7. Let G be a finite group with a split BN-pair of characteristic p, let V be a p-subgroup of G. Then, the following conditions are equivalent:

(i) (ii)

Mo( V)/V has a p-block of defect 0. V is G-conjugated to a Us, SC R.

Proof: The implication “(ii)+(i)” comes from the fact that -vG(U,) = L,, which is a group with a split BN-pair of characteristic p, hence has a p-block of defect 0: if k is any algebraically closed field of . a kL,-module that is projective (B.10.3, characteristic p, M,( 1, S) is Corollary 4) and irreducible (see also B.7.2.3). Conversely, let V be a p-subgroup of G such that .RJ V) has a p-block of defect 0. Up to G-conjugation one may suppose VC U. There exists a k&J V)-module rt which is irreducible and projective, for k an algebraically closed field of characteristic p. Let us consider ‘IL as a kJI/;,( V)-module on which V acts trivially. One has Hom,,ccy,(ind:“l;~~V,‘,k, n) # 0 since it equals, by Frobenius reciprocity, Hom,,-u( ,,,(k, res-,euu, ,,) n) = n’ “1”“1, which is non-zero since -vu(V) is a p-group. Hence there exists a short exact sequence with last terms ind:::;‘,Y,‘,k, rc, and 0; the morphisms are kJlr,( V)-morphisms but they are also kA’&( V)-morphisms since V acts trivially on ind:::$Y,‘,k. Since x is projective as &&( V)-module, the exact sequence splits: 7~)ind:z{Yy\ k. But in turn ind:::$~~ k( Y(V) ( see A.3), hence rc) Y(V) as kXG( V)-modules. Then, there exists a direct summand Y(x, S) of Y such that rc1Y(x, S)(V). Theorem 2 in A.4 now implies that V is a vertex of Y(x, S), hence (Proposition 6 abovej V is G-conjugate to U,. The proof is complete.

28

MARC CABANES

Remark. The interest of the above proof is to present a characterization of p-subgroups V such that sG( V) has p-defect 0 (a problem of importance in the framework of “local representation theory”) for these groups, the characterization being free of any consideration on reductive groups. The result however is not new: one could see it as a consequence of a stronger assertion due to Bore1 and Tits (see Cl]), which is the following (G is a finite group of Lie type): U,(NG( V)) = Vo V=, U, for an SC R (see also [6]).

But the result of Bore1 and Tits was proved using a process, due to Platonov, in reductive groups (see [ 19, Sect. 30.33). C.12.3. A general conjecture (due to J. L. Alperin) about “local representation theory” is that for any finite group G and any algebraically closed field of positive characteristic p, the number of G-conjugacy classes of pairs (V, x), where V is a p-subgroup of G and z an irreducible projective k&( V)-module (or a block of defect zero in kJ>( V)), equals the number of irreducible modules for kG. The verification of this conjecture for G a symmetric group and p any prime was announced by Alperin (August 1984). The following proposition shows it is also true for G a split &V-pair of characteristic p and k of the same characteristic. PROPOSITION 8. Zf G is a finite group with a split BN-pair of characteristic p, and if k is an algebraically closedfield of characteristic p, there are as many (isomorphism classes of) irreducible kG-modules as G-conjugacy classes of pairs (V, n), where V must be a p-subgroup of G and rc an (isomorphism class of) irreducible projective k2o( V)-module.

Proof: By the results of Part B, the irreducible kG-modules are parametrized by admissible pairs (x, S). If (x, S) is any admissible pair, let be the pair (U,, M,(x, S)). It is a pair (V, x) since M,(x, S) is an irreducible projective module for L, = JVG(U,). Moreover, any pair (V, TC) is G-conjugate to a pair (U,, M,(x, S)) for (x, S) an admissible pair: by Proposition 7, one may assume that V= U, for S c R, then 7cmust be an irreducible projective kL,-module. By B.lO, Corollary 4, there exists x E X,(T) such that R(X) I) S and ~2: M,(x, S). The proof will be complete when we have checked that (U,, M,(x, S)) is G-conjugate to (Us, M.&‘, s’)) if, and only if, (x, S) = (x’, s’). If gE G is such that U,=gUs then gEF(Usr U) so (Lemma 1 of B.8) US= Us, hence S=s’. Then M,(x, S) N M,(x’, S) so x = x’.

29

BRAUER MORPHISM

c.13. ml, mu,)

Our last theorem consists in computing the M-module Z( O,(P)) (notation of A.1.l ) for any direct summand Z of Y = indE k and any parabolic P. THEOREM 9. Let G be a finite group with a split BN-pair of characteristic p, let k be an algebraically closed field of characteristic p, let (x, S) be an admissible pair for G, and let JC R; then:

= Y,(x, S) as kP,module,

zj”S c J.

Proof: Y(x, S) is an indecomposable p-permutation kG-module with vertex U,. By [3, 3.2.(l)], one knows that Y(x, S)(V) #O if, and only if, VcG U,. But U,c, Us if, and only if, U,C U, (F( U,, U) = P, by B.8.1, Lemma 1) and U, c Us if, and only if, S c J. So Y(x, S)( U,) = 0 if, and only if, S d J. Now let JC R. We have seen that Y(U,) N Y,(L,, UL,) as klrmodules (B.8.1). We also know that Y(U,)= esCJ Y(x, S)(U,), the sum being taken over the set of admissible pairs (x, S) such that SC J; each term is non-zero. But the width of Y( U,) = Y(L,, U n L,) as kl/module is well known: it is the number of admissible pairs for L,, which is also precisely the number of terms in the above direct sum. Hence, if SC J and (1, S) is an admissible pair for G, Y(x, S)( U,) is an indecomposable direct summand of Y(L,, U n L,); therefore there exists an admissible pair for L,, (A, I), where ,JE X,(T) and Ic J n R(A), such that Y(x, S)( U,) = Y,(A, I). In order to prove the theorem, one must check that A = x and I= S. The composition map tiJ(l, I) 0Br”, gives a one-dimensional irreducible representation of J’$. This representation does not kill the idempotent i E Xk associated to Y(x, S) : Y,(& I) = Y(x, S)( U,) = (iY)( U,) = Br,,(i) . Y( U,) by A.2.3, Remark 2. Hence $,(A, I) 0Br,, since it does not kill i, in fact equals 1+9(x, S). But tiJ(ll, Z)oBr,, is clearly $(A, I) from the explicit formula for Bru, (see B.8.2, Theorem 6) and the definition of the notation $( .,.). So ;1= x and Z= S as required. The proof is now complete.

As a corollary we derive the Green correspondents of all the indecomposable summands of Y = indG,k. If Z is an indecomposable summand of Y, if H is a subgroup of G such that H 3 Ju,( V) for V a vertex of Z, one has resHZ = f( Z) 0 Z’, where Z’ is a sum of indecomposable kH-modules with vertices in the family of subgroups {A c H; A c H n V” for some XEG\H} (see [13, p. 1161) and f(Z) with vertex V.

30

MARCCABANES

Here the indecomposable kc-module Z is under the form Y(x, S) for (x, S) an admissible pair of G, and we know that a vertex of Z is V= Us (Proposition 6) HI NJ U,) = P,, hence H = Ps for some S c S’ c R. Then: COROLLARY 5. f( Y(x, S)) = Y&,

S) as kP,.-module.

Proof. Since f(Z), like Z, is of vertex I/= U, containing U,, and of source 1, since also Us a H, U,. acts trivially on f(Z), whence f(Z)( U,.) = f(Z) as kH-module. But f(Z) (resH Z, thus f(Z) ( Z( U,.); on the other hand Z( U,) = Y(x, S)( U,) is the indecomposable kP,-module Yr(x, S) by Theorem 9 above, which implies the equality of the corollary.

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19. J. HUMPHREYS,“Linear Algebraic Groups,” Graduate Texts in Mathematics, Vol. 21, Springer-Verlag. Berlin/New York/Heidelberg, 1975. 20. J. HUMPHREY&Defect groups for finite groups of Lie type, Math. Z. 119 (1971). 21. H. MATSUMOTO, Genkrateurs et relations des groupes de Weyl gtn&ralists, C. R. Acad. Sci. Paris 258 (1964), 3419-3422. 22. T. NAKAYAMA,On Frohenusean algebras, Ann. of Math. 40 (1939). 23. L. PUG, Pointed groups and constructions of characters, Math. Z. 176 ( 1981). 24. F. RICHEN, Modular representations of split BN-pairs, Trans. Amer. Marh. Sot. 140 (1969).

25. H. SAWADA,A characterization of the modular representations of finite groups with split &V-pairs, Math. Z. 155 (1977). 26. L. SCOTT.Modular permutation representations, Trans. Amer. Math. Sot. 175 (1973). 27. R. STEINBERG, Representations of algebraic groups, Nagoya Math. J. 22 (1963). 33-56. 28. N. TINBERG, Modular representations of linite groups with insaturated split RN-pairs, Canad. J. Marh. 32, No. 3 (1980).

29. N. TINBERG,Some indecomposable modules of groups with split BN-pairs. J. Algebra 61 (1979). 30. N. TINBERG,The Levi decomposition of a split BN-pair. Pacific J. Mufh. 91( 1) (1980).

4x1’115 l-3